Every group-embeddable monoid arises as the bimorphism monoid of some graph

TL;DR

The paper proves that every group-embeddable monoid $M$ is isomorphic to the bimorphism monoid Bi$(\Gamma^*)$ of some graph $\Gamma^*$, generalizing Frucht's finite-case automorphism realizations and the infinite de Groot–Sabidussi results from automorphisms to bimorphisms. The construction combines a bottom-layer modified Cayley graph with bimorphism-rigid gadgets to fix Aut and Bi on the base, and a top layer that encodes $M$ as a submonoid $B \cong M$, so that every bimorphism of the final graph corresponds to left-multiplication by an element of $B$. This establishes Bi$(\Gamma^*) \cong M$, clarifying the relationship between cancellativity and bimorphism realizations and providing a universal graph-theoretic realization for group-embeddable monoids. The approach also addresses which cancellative monoids can appear as epimorphism or monomorphism monoids, linking classical endomorphism-type questions to bimorphism representations.

Abstract

Generalizing results of Frucht and de Groot/Sabidussi, we demonstrate that every group-embeddable monoid is isomorphic to the bimorphism monoid of some graph.

Figures (6)

• Figure 1: Cayley graph of $M = (\mathbb{N},+)$ with respect to $A = \{1\}$.
• Figure 2: Graph $\Gamma$ obtained from the Cayley graph of $M = (\mathbb{N}_0,+)$ with respect to $A = \{1\}$.
• Figure 3: Construction of $\Gamma$ as in \ref{['urexample']}.
• Figure 4: Construction of the tree $\mathcal{G}$ in \ref{['brigid']}.
• Figure 5: A snippet of the graph $\Gamma$ constructed in \ref{['cayley']}. Vertices in $(g,R_a)$ are highlighted in red; vertices in $(ga,R_{a^{-1}})$ are highlighted in blue.

Theorems & Definitions (19)

• Theorem 1.1
• Example 1
• Example 2: coleman2017automorphisms
• Claim
• proof : Proof of claim
• Lemma 1
• proof
• Remark
• Remark
• Lemma 2